Center and Spread

Questions we are addressing in this section

• what exactly does the mean measure?

• How close to the mean are most of the data?

• How is the sample size related to the variability of the sample mean?

• How do empirical distributions of random sample means come out bell shaped?

• ho can we use sample means effectively for inference?

Definition of Mean and Average

The average or mean of a collection of numbers is the sum of all the element of the collection, divided by the number of elements in the collection

The methods np.average and np.mean

The Mean is a “Smooter”

You can think of taking the mean as an “equalizing” or “shooting” operation.

For example, imagine the entries in not_symmetric above as the dollars in the pockets of four different people. To get the mean, you first put all of the money into one big pot and then divide it evenly among the four people. They had started out with different amounts of money in their pockets ($2, $3, $3, and $9), but now each person has $4.25, the mean amount.

Proportions are Means

if a collection consists only of ones and zeroes, then the sum of the collection is the number of ones in it, and the mean of the collection is the proportion of ones.

ex:

zero_one = make_array(1,1,1,0)

sum(zero_one)

Output: 3

np.mean(zero_one)

The mean and the Histogram

The mean of the collection {2,3,,3,9} is 4.25, which is not the “halfway point” of the data. So then, what does the mean measure?

The mean of a collection depends only on the distribution of values in the collection. Therefore, if two collections have the same distribution, then they have the same mean

Here is an example:

non_symetric = {2,3,3,9}

mean = 4.25

same_distribution = make_array (2,2,3,3,3,3,9,9)

mean = 4.5

Mean is…

The mean is the center of gravity or balance point of the histogram

In general, for symmetry distributions…

the mean and the median are equal

What if the distribution is not symmetric